text: Because the carbonyl group is unable to serve a source of electrons for -M groups (in contrast to lone pair donors like OH), for reactions involving phenol and aniline starting materials, the σ values for electron-withdrawing groups will appear too small.  For reactions where resonance effects are expected to have a major impact, a modified parameter, and a modified set of σ constants may give a better fit.  This parameter is defined using the ionization constants of para substituted phenols, via a scaling factor to match up the values of σ with those of σ for "non-anomalous" substituents, so as to maintain comparable ρ values: for ArOH ⇄ ArO + H, we define .
  Likewise, the carbonyl carbon of a benzoic acid is at a nodal position and unable to serve as a sink for +M groups (in contrast to a carbocation at the benzylic position).  Thus for reactions involving carbocations at the α-position, the σ values for electron-donating groups will appear insufficiently negative.  Based on similar considerations, a set of σ constants give better fit for reactions involving electron-donating groups at the para position and the formation of a carbocation at the benzylic site.  The σ are based on the rate constants of the S1 reaction of cumyl chlorides in 90% acetone/water: for , we define .  Note that the scaling factor is negative, since an electron-donating group speeds up the reaction.  For a reaction whose Hammett plot is being constructed, these alternative Hammett constants may need to be tested to see if a better linearity could be obtained.